All Questions
69 questions
4
votes
0
answers
145
views
Indecomposable modules for the big quantum group
I am study the representation theory of the big quantum group at a root of unity, and I am wonder if it is known a complete classification of the indecomposable modules for it. To be more specific, ...
- 41
4
votes
0
answers
203
views
The Killing form on quantized enveloping algebras and reduction to the classical case
Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
- 3,637
3
votes
1
answer
367
views
The adjoint representation of $U_q({\frak sl}_2)$ on itself
Let $U_q(\frak{sl}_2)$ denote the quantum universal enveloping algebra of $\frak{sl}_2$, and consider the adjoint action
$$
\mathrm{ad}_X: U_q({\frak sl}_2) \to U_q({\frak sl}_2), ~~ Y \mapsto S(X_{(...
- 1,144
3
votes
1
answer
591
views
"Quantum Littlewood-Richardson" Rule?
Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...
3
votes
1
answer
492
views
Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]
Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as
$\operatorname{Rep}(G)$ ...
- 309
3
votes
1
answer
104
views
Irreducibility of product bicomodules
Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right,
$H$-comodule respectively. The tensor product
$$
V \otimes W
$$
has an obvious $H$-$H$-bicomodule structure.
If $V$ and $W$ are ...
- 1,144
3
votes
1
answer
190
views
Quantum group associated to a reductive group
In most of the classical references about quantum groups, these objects are defined as a one-parameter deformation of the universal enveloping algebra. However, I have read in several papers that it ...
- 31
3
votes
0
answers
97
views
Simple modules of quantum toroidal algebras
Many properties of quantum toroidal algebras are similar to quantum affine algebras. Every simple module of a quantum affine algebra of rank $n$ corresponds to an $n$-tuple of Drinfeld polynomials.
...
- 6,201
3
votes
0
answers
323
views
Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types
Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible finite-...
- 1,453
2
votes
1
answer
176
views
How does an element $T\left(z\right)$ act on a $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)\left[\left[z\right]\right]$-module?
Context
Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the ...
- 357
2
votes
0
answers
132
views
A question about q-binomials at roots of unity
I have a question about a lemma $9.3.6$ in the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. This question comes from page 301, "The restricted specialization&...
- 137
2
votes
0
answers
108
views
Why are the quantum Fock spaces in FLOTW the same as Uglov's?
Theorem 2.5 in the well-known FLOTW paper [1] and Theorem 2.1 in Uglov's paper [2] both refer to the original JMMO paper [3]
to define quantum Fock spaces, i.e. Fock spaces for $U_q(\widehat{\...
2
votes
0
answers
71
views
Comodules of the $B,C$ and $D$ series quantum groups
In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...
- 101
1
vote
1
answer
437
views
The coproducts $\mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ and $\mathbb{C}[U] \to \mathbb{C}[U] \otimes \mathbb{C}[U]$
A coproduct $\varphi: \mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ is given by: $x \mapsto 1 \otimes x + x \otimes 1$, where $x$ is a generator of $\mathbb{C}_q[U]$. There is a ...
- 6,201
1
vote
0
answers
166
views
How to understand a definition in KLR algebra in the setting of quantum affine algebras?
I am trying to figure out what should the following definition correspond to in the setting of quantum affine algebra:
$$
X \circ Y = Ind_{\beta, \gamma}^{\beta+\gamma} X \boxtimes Y \quad (1)
$$
This ...
- 6,201
1
vote
0
answers
88
views
Reference request: Nichols algebras of a braided vector space with a diagonal braiding
Are there some references of the proof of the following result?
Let $(V, c)$ be a braided vector space over a field $k$ with a basis $x_1, \ldots, x_n$, where $c$ is a diagonal braiding given by
\...
- 6,201
1
vote
0
answers
228
views
Clebsch Gordan coefficients of compact quantum groups
Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
- 11
1
vote
0
answers
216
views
polynomial representation of $sl_{2}(k)$
Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
\...
0
votes
0
answers
106
views
Hopf algebra antipodes and right left comodule equivalences
Given a Hopf algebra $H$, denote by ${}^H\mathrm{mod}$ the category of left $H$-comodules, and by $\mathrm{mod}^H$ the category of right $H$-comodules. If the antipode $S$ of $H$ is invertible then we ...
- 1,144